The potential influence of the ligament of Wieger on the crystalline lens shape

This research uses mathematical modelling to evaluate the influence of the ligament of Wieger on the crystalline lens shape at rest, and during accommodation. An axisymmetric model of the anterior segment, including the ligament of Wieger, was created using the finite element method. Different conditions including variations of stiffness and positions of the ligament, with and without the ligament, were tested to see how they affected lens curvature and optical power. Adding the ligament of Wieger to the simulation had a noticeable impact on the optical power of the lens, particularly on the posterior surface power and total power. Ligament stiffness and width significant influenced the accommodative range of the eye by − 0.95D and − 2.39D for ligaments with the same and 3× the stiffness of the capsular bag, respectively. Ligament width and inner diameter had negligible effects on lens thickness but did have significant effects on posterior surface power and accommodation. In this simulation, we found that the ligament of Wieger can significantly affect the lens shape, both at rest and during accommodation, and may need to be considered in lens models.


Method Geometry
An axisymmetric FEM model of the entire eye was developed in ANSYS Mechanical 2023R2 (Ansys Inc, Southpointe, PA, USA), which included the cornea, limbus, sclera, iris, zonular fibers, ciliary muscles, crystalline lens, lens capsule, and ligament of Wieger, which aligns with previously published models for a 29-year-old eye 5,6,13 .Similar to Ljubimova 11 , the ligament of Wieger was assumed to follow the shape of the posterior surface of the lens.The lenticular cortex and nucleus are considered separately, as they both contribute to the lens overall power and shape, and their inclusion can provide more detailed insight into the lens reshaping 4,14 .
The cornea, sclera, iris, ciliary muscles, and crystalline lens were modelled in the accommodated state, including the cortex and nucleus (Table 1), while the lens capsule was considered as a surface coating of 10 µm in thickness 6 .The zonular fibers were defined as five sets of link elements with a diameter of 40 µm that connect the ciliary muscles and the lens to transfer the tension applied by the ciliary body (Fig. 1).As the precise anatomical location of the ligament remains to be conclusively identified, the proposed location was based on anatomical descriptions from literature. 10,15Therefore, the ligament of Wieger was modeled as a 20 µm thick ring-shaped membrane at three specific positions: the first and second had an inner diameter of 6 mm and a width of 0.5 mm or 1 mm respectively, while the third had an inner diameter of 7 mm and a width of 0.5 mm (Fig. 2).The anterior and vitreous chambers were not considered in the model to simplify the analysis and decrease the computational cost.

Material properties
Material properties were assigned to each structure based on literature values (Table 2).The first-order Ogden hyperelastic model was used to model the material behaviour of the cornea.The strain energy potential of this model is derived from the principal components of the left Cauchy-Green deformation tensor and is expressed as follows: where μ i and α i are material constants and D k is the incompressible parameter indicating volume changes.J is the volume ratio before and after deformation and for incompressible materials J = 1.The sclera and limbus were characterized using the Yeoh third order hyperelastic model, which accounts for the non-linear response of materials to different loading and strain conditions.The model is described as follows: Here, I 1 represents the first invariant of the Cauchy-Green strain tensor, and C i0 denotes the shear modulus that correlates with sclera behaviour.
Materials with elastic properties were chosen for the other components of the eye model.The response of these materials is commonly described by Hooke's law, which in its basic form is expressed as σ = E⋅ϵ, where σ is the stress, E is Young's modulus-a measure of the stiffness of the material-and ϵ is the strain, the deformation of the material relative to the applied stress.

Mesh and boundary conditions
Considering the symmetry of the lens about its central axis and the negligible effect by the posterior segment of the eye on accommodation, it was deemed sufficient to simulate only an axisymmetric model of the anterior eye while significantly reducing computational costs (Fig. 1).
The model featured an axisymmetric triangular mesh, utilizing the PLANE183 element type for the cornea, sclera, limbus, iris, ciliary body, lens cortex, and nucleus, and all the connections between these components were bonded type.The SHELL208 element type was selected for the lens capsule, which was an axisymmetric shell as a surface coating for the cortex.Finally, the zonular fibers were meshed using the LINK180 element type with circular cross section.Ansys automatically generated this configuration, ensuring that each element type was compatible with the axisymmetric nature of the model.
To ensure appropriate modelling, we restricted the bottom of the sclera to prevent unwanted translations or rotations of the eye.We restricted movement of the cornea and lens to solely the optical axis.Regarding the Wieger ligament, we treated it as a component bonded to the lens capsule without adding any further boundary conditions (Fig. 3).

Mesh convergence analysis
A mesh sensitivity analysis was conducted to determine the optimal mesh size.Therefore, five models were generated, each with a different mesh density (Table 3).Our analysis revealed that the displacements at the equator, anterior pole, and posterior pole of the lens converged for the model containing 7658 elements and 3420 nodes (Fig. 4).Consequently, we selected this mesh density for the formal analysis in our study.

Simulation
Accommodation is controlled by the ciliary muscle contraction, which was simulated by displacing the muscle 0.5 mm away from the central axis (Fig. 3) in 8 steps, indicated as 'levels of zonular stretching' between 0 and 100%.This stretching was preferred over the accommodation as a reference since the presence of the ligament would affect the amount of accommodation that can be accomplished.Starting from the fully accommodated state with a power of 29.65D, zonular stretching gradually leads to the flat lenticular shape of the non-accommodated state.The effect of the ligament of Wieger on lens shape and power was investigated by repeating this www.nature.com/scientificreports/process under several conditions.The first analysis was performed without the ligament to serve as a control, followed by four analyses with a ligament at different stiffness levels (half, once, twice, and three times the stiffness of the lens capsule) with a width of 0.5 mm and an inner diameter of 6 mm (marked as '0.5W6D').Two additional analyses were conducted to evaluate the impact of ligament width and inner diameter.The first involved a ligament with a width of 1 mm and an inner diameter of 6 mm ('1W6D'), while the second used ligaments with a width of 0.5 mm and an inner diameter of 7 mm ('0.5W7D').
The nodal coordinates after deformation were extracted from the anterior and posterior lens surface within 5 mm diameter in optical zone from the model for each analysis and used to determine the lens surface radii of curvature ( r a and r p ) by post analysis in MATLAB (R2023a, The Mathworks, Natick, MA, USA) and Excel (v365, Microsoft, Seattle, WA, USA).To do this, both lens surfaces were fitted by an aspherical surface given by: where x is the radial distance from the optical axis and a is curvature and b is conic constant.These values were used to calculate the anterior, posterior, and total power with the following formulas: with LT, the lens thickness (in meter), r a and r p the lens radii of curvature (in meter), n l = 1.43, the equivalent lens refractive index, and n a = 1.336, the refractive index of the aqueous humor.In practice, the refractive index of 1.42 given by Burd et al. 14 was found to give lens power values that were unrealistically low.Hence, the refractive index from the SyntEyes model 23 was used instead.
The total force exerted by the zonular fibers was determined using ANSYS by extracting the forces of each zonular fiber directly and then adding these values to obtain the overall zonular force.

Results
On a macroscopic scale, the influence of the ligament of Wieger on the non-accommodated lens shape appears modest, but the optical consequences would be noticeable. (3)

Influence of ligament stiffness
For a ligament with the same stiffness as the lens capsule, the non-accommodated lens thickness increased by 38 μm and by 97 μm for the highest stiffness (Fig. 5a).Lens diameter and the anterior surface power saw only minor changes for the highest level of ligament stiffness with increases of 0.007 mm and 0.1D, respectively, (Fig. 5b,c).As expected, the non-accommodated posterior surface power is most affected by the ligament, with increases of 0.92D and 2.34D for ligaments with the same and three times the stiffness of the capsular bag (Fig. 5d).Very similar values are found for the changes in total lens power (Fig. 5e).Since the accommodative range can be defined as the difference in lens power between the fully accommodated and non-accommodated states, an increase in the latter while the former remains constant will inevitably affect the range by − 0.95D and − 2.39D for ligaments with the same and three times the stiffness of the capsular bag (Fig. 5f).

Influence of ligament width and inner diameter
To assess the influence of the width and inner diameter model 0.5W6D with the same stiffness as the capsular bag was considered as a reference.Increasing the inner diameter to 7 mm (0.5W7D) had only a negligible effect on the non-accommodated less thickness states, while a broader ligament led to a minor increase in thickness by 22 µm (Fig. 6a).There was, however, a noticeable difference in posterior surface power, with changes of + 0.19D for 1W6D and − 0.33D for 0.5W7D (Fig. 6b).The corresponding changes in accommodation were − 0.26D and + 0.37D, respectively (Fig. 6c,d).

Zonular forces
At maximum accommodation the total force exerted by the zonular fibers was 0.064 N (Fig. 7).This is in close agreement with the findings of a study using a 29-year-old lens model, where the average total net force from the zonular fibers and ciliary muscle was reported as 0.056 N 24 .Additionally, our result aligns with the force of 0.055 N recorded in an in vitro stretching experiment on a 47-year-old human eye 25 , further corroborating the reliability and relevance of our simulation data.

Discussion
Not much is known about the ligament of Wieger is a difficult structure to observe and measure in vivo.Under certain conditions, however, such as in pigment dispersion syndrome, where Scheie's line may appear [26][27][28] , trauma 29 , or intraocular hemorrhage 30 the normally transparent structure may become visible.The exact function of the ligament is not fully understood.Initially, it was believed to be a vestigial remnant of the fetal hyaloid vasculature, but it has since been suggested that it may play a role in accommodation 11 .Building on this, Goldberg theorized that the ligament of Wieger cradles the lens to hold it in place using additional zonules, although the existence of these zonules is yet to be confirmed.Our simulations show that the presence of the ligament, attached to the posterior lens surface, affects both the shape of the non-accommodated lens and the accommodative range in the model (Fig. 5).These effects become more important for higher levels of ligament stiffness, although the actual stiffness remains unknown.The width and inner diameter of the ligament of Wieger also have a smaller, but noticeable impact on the optical properties of the lens.These findings are consistent with those observed by Ljubimova et al. 11 , who suggested that the inclusion of the ligament leads to smaller changes in the posterior radius of curvature, indicating increased rigidity in the posterior lens surface.Moreover, our results demonstrate that changes in central thickness and curvatures with accommodation also reduced in the presence of the ligament compared to when it was not included.
Another important aspect to consider is the strength of the adhesion between the ligament and the capsular bag.This is thought to be strong in young eyes, but gradually weakens with age until the ligament detaches, as is seen clinically in many individuals over the age of 65 years 31 .In this model, such a detachment would appear as a decrease in posterior capsular stiffness.Given that the posterior lens curvature does not change (much) during adulthood 32 , as opposed to the anterior curvature that sees large changes, one may speculate that the presence of the ligament preserves the posterior lens shape.But since the crystalline lens fibers never stop dividing 33 , thus increasing the thickness and curvature of the lens, it is conceivable that stress gradually increases at the area where the ligament is attached to the capsule, leading to their eventual separation.This gradual detachment reduces the ligament's influence on the posterior lens shape, allowing it to deform more during accommodation, which may temporarily dampen the effects of presbyopia.This is speculative, however, until further in vivo data can be obtained.These findings highlight the importance of further investigating the role of the ligament of Wieger in age-related changes in the lens and may have implications for the development of future treatments for presbyopia.
As with any model, it is important to note that it is based on several assumptions to approximate the structure of the eye, which may have had some minor consequences for the model's accuracy.One such assumption was that the ligament was attached in the fully relaxed state.Alternatively, one could attach it in the fully stretched state, but then the lens would not be able to reach the same level of full accommodation.When attached in an in-between state, it would affect both the relaxed and accommodated states, influencing the range of accommodation even further.Furthermore, the model assumes that the accommodation works in reverse, from a relaxed to a non-relaxed state, but this should not affect the results.Age could be a factor as well since some studies 34,35 indicate that in young lenses the shear modulus of the nucleus could be as high as or higher than that of the cortex, which could affect the model's predictions.Other possible factors include the variations in capsular thickness into the model, which could have slightly alter the results, or conditions affecting the adhesion of the Wieger ligament, such as diabetes 36 .Note also that the mechanical influences of the ligament presented in this work greatly depend on its unknown stiffness.This was addressed by considering a broad range of stiffness values    In conclusion, the results of this analysis suggest that ligament of Wieger has a noticeable effect on lens shape and accommodation that is important enough to merit consideration when biomechanically modelling these processes.

Figure 1 .
Figure 1.Schematic of the lens geometry; inset: Insertion point of the ligament of Wieger.

Figure 2 :
Figure 2:.3D schematic of the posterior lens surface.Left: ligament of 0.5 mm or 1 mm width and a 6 mm inner diameter; right: ligament of 0.5 mm width and a 7 mm inner diameter.

Figure 4 .
Figure 4. Graphical representation of displacement at the anterior apex, posterior apex, and equatorial region across different mesh densities.A mesh configuration with 3420 nodes and 7658 elements has been determined to be the minimum required to ensure convergence of the computational model, indicating the point at which further refinement does not result in significant changes to displacement outcomes.
https://doi.org/10.1038/s41598-024-54674-wwww.nature.com/scientificreports/ that likely includes that correct value, as well as values that are too high or too low.Finally, the model does not consider the effects of fluidic transmitter forces that would have slightly dampened accommodation.

Figure 5 .
Figure 5. Changes in parameters describing the shape of the crystalline lens for various accommodative states, ranging between full accommodation (0% zonular stretching) and no accommodation (100% zonular stretching), and various values of ligament stiffness (in multiples of capsular bag stiffness).(a) Lens thickness; (b) Lens diameter; (c) Anterior surface power; (d) Posterior surface power; (e) Total lens power; (f) Accommodation.

Figure 6 .
Figure 6.Changes in parameters describing the shape of the crystalline lens.

Figure 7 .
Figure 7. Zonular forces in the model without the ligament of Wieger.

Table 1 .
Geometrical parameters for eye components in mm.

Table 2 .
Material properties for eye components.

Table 3 .
P total = P anterior + P posterior − P anterior • P posterior • Mesh analysis.